How do you simplify sin x cos (x+y) + cos x sin (x+y)?

Answer 1

Let x + y = a
sin x cos a + cos x sin a
= sin (x + a)
= sin (x + x + y)
= sin (2x + y)

Answer 2

note that sin(a+b) = sina * cosb + cosa * sinb
let a = x, b = x+y

so equation equals sin(2x+y)

Answer 3

sin(x)cos(x + y) + cos(x)sin(x + y)
sin(x)(cos(x)cos(y) - sin(x)sin(y)) + cos(x)(sin(x)cos(y) + sin(y)cos(x))
sin(x)cos(x)cos(y) - sin^2(x)sin(y) + cos(x)sin(x)cos(y) + sin(y)cos^2(x)